Towards Quantifying Vertex Similarity in Networks

TL;DR

This paper introduces novel methods for quantifying vertex similarity within and across networks, using geometric optimization and Laplacian-based measures, supported by algorithms with proven effectiveness on synthetic and real data.

Contribution

It presents a new geometric approach for vertex similarity and a Laplacian-based graph matching measure, along with deterministic and randomized algorithms for network alignment.

Findings

High-quality vertex similarity results achieved

Effective graph matching on synthetic and real data

Provides insights into network structure

Abstract

Vertex similarity is a major problem in network science with a wide range of applications. In this work we provide novel perspectives on finding (dis)similar vertices within a network and across two networks with the same number of vertices (graph matching). With respect to the former problem, we propose to optimize a geometric objective which allows us to express each vertex uniquely as a convex combination of a few extreme types of vertices. Our method has the important advantage of supporting efficiently several types of queries such as "which other vertices are most similar to this vertex?" by the use of the appropriate data structures and of mining interesting patterns in the network. With respect to the latter problem (graph matching), we propose the generalized condition number --a quantity widely used in numerical analysis-- $\kappa(L_G,L_H)$ of the Laplacian matrix representations of $G,H$ as a measure of graph similarity, where $G,H$ are the graphs of interest. We show that this objective has a solid theoretical basis and propose a deterministic and a randomized graph alignment algorithm. We evaluate our algorithms on both synthetic and real data. We observe that our proposed methods achieve high-quality results and provide us with significant insights into the network structure.